{"id":"e7be37fc-5f17-4c90-b8b5-37f6ccbd66d7","arxiv_id":"2506.07176","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For non-symmetric convolution-type operators with periodic coefficients, the resolvent is approximated in operator norm by a homogenized diffusion resolvent with drift, with error O(ε).","lead":"This paper proves that a non-symmetric, nonlocal convolution-type operator with periodic microstructure is well approximated, in operator norm, by a homogenized diffusion operator with a drift term, with an error of order epsilon. The result extends the operator-theoretic homogenization method, previously limited to self-adjoint problems, to the non-self-adjoint setting.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The O(ε) estimate in Theorem 4.1 depends on strict positivity of q0 (Prop. 1.1(3)), whose proof relies on the convolution-power lower bound in Theorem 6.1; if that lemma fails for general L1 kernels, the lower bound (1.39) and all resolvent estimates collapse.","rationale":"The paper's central theorem is a substantial extension of the Birman-Suslina method to non-self-adjoint nonlocal operators. I went through the main chain: direct integral representation, spectral analysis of A(ξ), threshold expansions via contour integrals, resolvent approximation in each fiber, and the final scaling argument. The algebraic identities used in Proposition 3.1 and Theorem 3.2 check out, including the decomposition (3.18)-(3.19) of Ξ. The scaling relations in Theorem 4.1 are correct. The proof of positivity of q0 is the least secure link: it is deferred to Theorem 6.1 and depends on a cited convolution-power lemma that is not re-proved. Everything else, including the effective-drift term and the positivity of g0, follows from the two-sided bounds on q0. Thus the reader's identified weakest assumption is indeed the load-bearing one. The 'sharp in order' claim lacks an explicit lower bound, but since the symmetric case is a special case and would already give optimality if a lower bound is known, this is a presentation issue rather than a correctness risk. The minor identity in Concluding Remark 3 for strong convergence appears questionable, but it does not affect Theorem 4.1. No change to the reader's CONDITIONAL verdict is warranted.","tokens_in":34887,"tokens_out":45161,"duration_ms":434267,"concrete_test":"Independently verify Theorem 6.1 and [27, Lemma 4.2] under exactly (1.1)-(1.3): (1) Re-derive the lemma from scratch; try to construct a counterexample with a∈L1 nonnegative, positive measure, but supported on a fat Cantor-type set, or with an integrable |x|^{-α} singularity, and check whether some F_N admits a uniform positive lower bound. (2) Numerically solve the cell problem A(0)^* q0 = 0 on a fine grid for a non-symmetric (a, μ) with unbounded L1 kernel, and check min q0 > 0 with a margin and that (1.39) holds with q_- > 0. (3) For the same kernel compute F_N = ea^{*N} and find the smallest N with inf F_N > 0; confirm N is finite and independent of discretization.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central estimate is built on the two-sided bound 0 < q_- ≤ q0 ≤ q_+ used to prove accretivity of [q0^{1/2}]A(ξ)[q0^{-1/2}] (Corollaries 1.7, 1.9, 1.14), the key quadratic-form lower bound (1.39), and positive definiteness of g0 in Section 3.1. Positivity of q0 is established in Proposition 1.1 via Theorem 6.1, where the decisive step is Lemma 6.2/Lemma 6.4: they require existence of N and γ > 0 such that the convolution power F_N(z) ≥ γ for the periodization ea(z) = Σ_n a(z+n). This is cited from [27, Lemma 4.2] and not re-proved. For a merely L1 nonnegative kernel, convolution powers can be unbounded or have zeros unless the cited lemma supplies extra structure. If the lemma needs additional regularity (L∞, or support with nonempty interior) beyond (1.1)-(1.3), then q0 might fail to be uniformly positive, the similarity transform would not yield the resolvent bounds (1.40), and the threshold approximations of Propositions 2.3-2.4 and Theorem 3.2 would not produce the O(ε) estimate in Theorem 4.1. This is the single most load-bearing external assumption in the proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies homogenization of non-symmetric convolution type operators of the form (0.1) in L2(Rd). Under assumptions (1.1)–(1.3) and the third moment condition M3(a)<∞, Theorem 4.1 asserts an O(ε) operator-norm approximation of the resolvent (Aε+I)^{-1} by the operator (A0+ε^{-1}⟨α,∇⟩+I)^{-1} multiplied on the right by q0(x/ε), where A0=-div g0∇ is the effective elliptic operator and α is the effective drift. The proof combines the Gelfand transform, a reduction to the fiber operators A(ξ), threshold expansions of the Riesz projector, accretivity estimates for [q0^{1/2}]A(ξ)[q0^{-1/2}], and cell problems for the effective coefficients. Sections 5 and 6 contain supporting material on stability of isolated eigenvalues and on the auxiliary operator G, including the proof that the adjoint kernel q0 is strictly positive and bounded.","tokens_in":35214,"tokens_out":13879,"duration_ms":154721,"significance":"If the main estimate is correct, this is a substantial extension of the Birman–Suslina operator-theoretic homogenization method to non-self-adjoint, nonlocal convolution-type operators. The explicit cell problems for α and g0, the moving-frame interpretation of the ε^{-1} drift term, the absence of fitted parameters, and the operator-norm nature of the error estimate are genuine strengths. The result is conditional on the positivity of q0, which in turn depends on a convolution-power lower bound quoted from a previous work; this point needs to be clarified before the significance can be fully certified.","major_comments":[{"comment":"The proof of Lemma 6.2 relies on the assertion, quoted from [27, Lemma 4.2], that there exist N and γ>0 such that the N-th convolution power F_N of the L1 periodization ea is bounded below by γ on Ω. This assertion is load-bearing: it is the only input that gives the strict positivity (1.10) of q0, which is then used in identity (1.15), Corollaries 1.7, 1.9 and 1.14, the quadratic-form lower bound (1.39), and the positive definiteness of g0 in Section 3.1. The manuscript neither states the hypotheses of [27, Lemma 4.2] nor proves the lower-bound statement under (1.1)–(1.3). For a general nonnegative L1 kernel with positive measure whose periodization has support with empty interior, convolution powers need not be uniformly positive on Ω. The authors should state the lemma in full and verify its assumptions under (1.1)–(1.3), or supply a self-contained proof; if an additional support or regularity condition is needed, the hypotheses of Theorem 4.1 must be modified accordingly.","section":"§6.2, Lemma 6.2; Prop. 1.1(3)"}],"minor_comments":[{"comment":"The heading contains the typos 'Introdiction' and 'Secrions'; 'six Secrions' should read 'six Sections'.","section":"§0.2"},{"comment":"There are two typos: 'conrour' should be 'contour', and in (2.39) the left-hand side is repeated as 'w_j(x) = w_j(x) ='.","section":"§2.2, proof of Prop. 2.4"},{"comment":"In the sentence 'for any n∈N' and in formula (6.7), the symbol should be N (the same letter as in G^N) rather than n.","section":"§6.2, Lemma 6.4"},{"comment":"The sentence 'From this inequality, taking into account (6.17)...' should refer to the almost-everywhere convergence established just before, so 'From this convergence' is more accurate.","section":"§6.6"},{"comment":"The phrase 'sharp in order' is stronger than what is proved: Theorem 4.1 establishes an O(ε) upper bound but no matching lower bound or example showing that the order cannot be improved. Please either add a lower-bound result or replace 'sharp in order' by 'with discrepancy of order O(ε)' throughout.","section":"Abstract and §4.1"}],"recommendation":"major_revision","confidential_remarks":"The essential risk is the unstated hypothesis in [27, Lemma 4.2] used in Lemma 6.2. If that lemma requires more than L1 nonnegativity with finite moments, the positivity of q0 and therefore the whole O(ε) estimate in Theorem 4.1 would not be guaranteed under the stated assumptions. I recommend asking the authors to reproduce the lemma or prove it in the appendix. The 'sharp in order' wording should also be toned down unless a lower bound is supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is the first operator-norm homogenization estimate for non-symmetric convolution-type operators, and the main theorem looks right. The approximation requires both an effective drift ε^{-1}⟨α,∇⟩ and multiplication by the periodic factor q0(x/ε) — that is genuinely new and not just a cosmetic addition. The effective coefficients come from cell problems, not from fitting, and the proof is careful and mostly self-contained.\n\nThe method is coherent: rescale, decompose into a direct integral, and use finite smoothness from the moment assumptions to do a threshold analysis near ξ=0. The quadratic-form identity in Lemma 1.6 is a neat trick, and the appendices on the auxiliary operator G are substantial and honest work.\n\nThe one point I would send a referee to check is the strict positivity of q0. That rests entirely on [27, Lemma 4.2], which asserts that some convolution power of the periodized L1 kernel is bounded below by a positive constant. The paper cites it without proof. If that lemma needs extra structure beyond (1.1)–(1.3) — say boundedness or a support with nonempty interior — the main estimate collapses. I did not find a counterexample, and the proof's use of the lower bound is not obviously excessive, but this is load-bearing and should be verified. The stress-test note flags exactly the right risk.\n\nAlso, the phrase 'sharp in order' is not backed by a matching lower bound. The O(ε) upper bound is proved, and the remark giving O(ε^{k-2}) for 2<k<3 is plausible, but 'sharp' is an assertion, not a result. Minor typos in Section 0.2 and some notational slips are harmless.\n\nIf you are in quantitative homogenization, this is worth a serious referee. The architecture is sound, the novelty is real, and the cited lemma is likely checkable. I would send it to review with a request to confirm [27, Lemma 4.2] (either prove it or give a full reference) and to soften the 'sharp' claim. My own verdict is conditional, leaning accept after that verification.","headline":"Solid non-self-adjoint extension of operator homogenization; verify the cited convolution-power lemma before accepting.","tokens_in":35716,"tokens_out":29980,"would_cite":true,"duration_ms":335241,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B27","45K05","47G10","47A55"],"pacs":[],"model":"deepseek-v4-flash","headline":"Non-symmetric convolution operators admit O(ε) resolvent homogenization.","keywords":["convolution type operators","periodic homogenization","operator estimates of discrepancy","effective operator","non-self-adjoint operators","effective drift","Gelfand transform","resolvent approximation"],"falsifier":"Take a concrete non-symmetric pair (a,µ), compute α and g0 from (2.40) and (2.47), and evaluate the L2→L2 norm of the difference (A_ε+I)^{-1} − ($A^{0}$+$ε^{{-1}}$⟨α,∇⟩+I)^{-1}[q_0^ε] numerically for ε = $2^{{-n}}$; if the norms do not stay below C ε for some fixed C, or if the bound fails when q0 is replaced by a sign-changing solution of the adjoint equation, the main theorem is false.","tokens_in":34719,"feed_emoji":"📐","tokens_out":6393,"duration_ms":55369,"temperature":0.7,"pith_summary":"The paper proves a sharp operator-norm approximation for a family of non-symmetric convolution type operators with periodic coefficients on L2(R^d). For small ε, the resolvent (A_ε+I)^{-1} is approximated to order O(ε) by the resolvent of a constant-coefficient elliptic operator with an added drift of order $ε^{{-1}}$, multiplied by a rapidly oscillating periodic factor q0(x/ε). The effective diffusion matrix and drift vector are computed from cell problems, and q0 is the positive kernel of the adjoint periodic operator, normalized to mean one. This extends the operator-theoretic homogenization approach from self-adjoint to non-self-adjoint nonlocal operators, and shows that the associated Cauchy problem homogenizes only in a moving frame.","feed_headline":"Non-symmetric jump operators homogenize with O(ε) error","feed_subtitle":"A periodic drift and diffusion, plus an oscillating factor, approximate the resolvent uniformly in ε.","key_machinery":"The load-bearing mechanism is the threshold approximation near the spectral edge of the fibered operator family A(ξ) obtained by the Gelfand transform. At ξ=0, the operator has an isolated simple eigenvalue λ0=0, with kernel spanned by constants and adjoint kernel spanned by a strictly positive periodic function q0. For small ξ, the Riesz projection F(ξ) and the product A(ξ)F(ξ) are expanded to order O(|ξ|) and O(|ξ|^3) respectively; the linear coefficient yields the effective drift i⟨α,ξ⟩P and the quadratic coefficient yields the effective matrix g0. A quadratic-form lower bound for Re([q0]A(ξ)) then controls the resolvent (A(ξ)+$ε^{2}$ I)^{-1} with precision O($ε^{{-1}}$), which after unitary scaling becomes the O(ε) resolvent estimate for A_ε.","core_discovery":"Under conditions (1.1)–(1.3) and the finite third moment M3(a)<∞, the paper establishes ‖(A_ε+I)^{-1} − ($A^{0}$ + $ε^{{-1}}$⟨α,∇⟩+I)^{-1}[q_0^ε]‖_{L2→L2} ≤ C ε for all ε>0, where $A^{0}$ = −div g0∇ is the effective diffusion operator with positive definite constant matrix g0, α is a constant vector called the effective drift, and [q_0^ε] is multiplication by the ε-periodic positive function q0(x/ε). The estimate is sharp in order, and the constant depends only on the listed data of a and µ. In particular, the resolvent of A_ε does not converge in the usual sense to the resolvent of a fixed limit operator; the leading approximation requires both the large drift $ε^{{-1}}$⟨α,∇⟩ and the oscillating factor q0. The proof combines a Gelfand-transform direct integral with threshold approximations of the spectral projection near the isolated eigenvalue zero of the fibered operator A(ξ).","pith_inferences":["The same threshold-expansion strategy should transfer to non-self-adjoint periodic differential operators, where the effective drift would enter as a first-order symbol term and the adjoint ground state would play the role of q0.","Because the approximation keeps the oscillating factor q0(x/ε), the leading-order asymptotics carries a two-scale structure; a single autonomous effective operator cannot describe the resolvent on L2(R^d).","For non-symmetric Lévy-type operators with stable-like kernels, an analogous operator-norm estimate with a drift term should hold, with q0 corresponding to the invariant density of the periodic process; this is a testable extension of the present method."],"forward_implications":["The resolvent estimate (4.1) holds uniformly for all ε>0 with a constant depending only on d0, K, q−, q+, d, µ±, M1(a), M2(a), M3(a), M(a), Cπ(a), and Cr(a)(a).","If only M2(a)<∞ is assumed, the same approximating operator still gives convergence as ε→0, but without a rate; if M_k(a)<∞ for some 2<k<3, the rate becomes O(ε^{k−2}).","For the parabolic Cauchy problem ∂_t u = −A_ε u, the result implies homogenization in the moving frame (x,t) ↦ (x − α t/ε, t); in the original frame the semigroup does not converge in the usual strong topology.","The statement extends to arbitrary periodic lattices in R^d, with constants depending on the lattice parameters."],"supporting_citations":[{"why":"Supplies the symmetric-case operator estimates and the contour-integration threshold technique that this paper adapts to non-symmetric operators.","marker":"[22]"},{"why":"Establishes existence, positivity, and normalization of the adjoint kernel q0 via the Krein–Rutman theorem, and homogenization in moving coordinates for biased convolution operators.","marker":"[27]"},{"why":"Introduces the operator-theoretic Gelfand-transform approach and O(ε) resolvent estimates for periodic elliptic operators that this paper generalizes.","marker":"[4]"},{"why":"Presents the symmetric-case analogue whose proof remains valid without symmetry, cited for the expansion lemma used here.","marker":"[23]"},{"why":"Provides the first homogenization result for symmetric convolution type operators, identifying the effective second-order elliptic operator.","marker":"[26]"},{"why":"Supplies the stability-of-isolated-eigenvalue perturbation result used to control the spectral projection near ξ=0.","marker":"[16]"}],"fun_headline_variants":["Non-symmetric jump operators homogenize with sharp ε error","Effective drift and diffusion for non-symmetric convolution operators","Sharp resolvent approximation in non-symmetric homogenization","O(ε) resolvent bound for non-symmetric jump-type homogenization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the periodic adjoint kernel q0 being strictly positive and bounded; if q0 could vanish on a set of positive measure, the quadratic-form lower bound and the O(ε) resolvent estimate would fail.","fun_headline_variants_meta":{"raw":{"variants":["Non-symmetric jump operators homogenize with sharp ε error","Effective drift and diffusion for non-symmetric convolution operators","Sharp resolvent approximation in non-symmetric homogenization","O(ε) resolvent bound for non-symmetric jump-type homogenization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000503,"raw_usage":{"total_tokens":2557,"prompt_tokens":1148,"completion_tokens":1409,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":764,"completion_tokens_details":{"reasoning_tokens":1340}},"tokens_in":764,"tokens_out":1409,"duration_ms":9818,"temperature":1.0,"reasoning_tokens":1340,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:40:14.012736+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete non-symmetric pair (a,µ), compute α and g0 from (2.40) and (2.47), and evaluate the L2→L2 norm of the difference (A_ε+I)^{-1} − ($A^{0}$+$ε^{{-1}}$⟨α,∇⟩+I)^{-1}[q_0^ε] numerically for ε = $2^{{-n}}$; if the norms do not stay below C ε for some fixed C, or if the bound fails when q0 is replaced by a sign-changing solution of the adjoint equation, the main theorem is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the symmetric-case operator estimates and the contour-integration threshold technique that this paper adapts to non-symmetric operators."},{"cited_title":"Asymptotic Anal","cited_arxiv_id":null,"evidence_quote":"Establishes existence, positivity, and normalization of the adjoint kernel q0 via the Krein–Rutman theorem, and homogenization in moving coordinates for biased convolution operators."},{"cited_title":"Sh., Suslina, T","cited_arxiv_id":null,"evidence_quote":"Introduces the operator-theoretic Gelfand-transform approach and O(ε) resolvent estimates for periodic elliptic operators that this paper generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Presents the symmetric-case analogue whose proof remains valid without symmetry, cited for the expansion lemma used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the first homogenization result for symmetric convolution type operators, identifying the effective second-order elliptic operator."},{"cited_title":"Springer-Verlag, Berlin (1980)","cited_arxiv_id":null,"evidence_quote":"Supplies the stability-of-isolated-eigenvalue perturbation result used to control the spectral projection near ξ=0."}],"review_version":1}